Analysis of a Reaction–Diffusion Cholera Model with Distinct Dispersal Rates in the Human Population

In this paper, we investigate the long term behaviors of a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition. The parameters in the model involving space are typical space-dependent due to spatial heterogeneity. We consider the case that the dispersal rates of the susceptible and infected hosts are different, no diffusion term in the cholera equation and bilinear incidence infection mechanism. The existence of global solution, uniform boundedness of solution, asymptotic smoothness of semiflows and existence of global attractor are also addressed. We define the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {R}_0$$\end{document} for the model for the disease transmission in spatially homogeneous environment and establish a threshold type result for the disease eradication or uniform persistence. Considering the cases that either the dispersal rate of the susceptible individuals or the dispersal rate of the infected individuals approaches zero, we investigate the asymptotical profiles of the endemic steady state. Our results suggest that: cholera can be eliminated by limiting the movement of the susceptible individuals, while limiting the mobility of the infected hosts, the infected individuals concentrate on certain points in some circumstances.